Differentiation Formula, Definition, Concept and Examples

Differentiation Formula, Definition, Concept, and Examples: In the study of calculus, Differentiation refers to the process of calculating the derivatives of a function. A derivative is defined as the rate of change of a particular function with regard to another function in mathematics or physics mainly. The eminent scientist, Isaac Newton, propounded the laws of Differential Calculus. The study of Differentiation and Integration makes the main concepts of calculus in mathematics. Many branches of science deal with the principles of limits and derivatives. In this article, you will know more about the definition of differentiation and derivatives, the rules of differentiation, and different formulas of differentiation to find the derivatives of algebraic functions, trigonometric functions, and exponential functions. These topics of differentiation are very useful for students of classes 11th and 12th as differentiation is mentioned in their syllabus.

What is Differentiation?

In calculus, the process of differentiation denotes the rate of change of one quantity with respect to another one. For example, the speed of a moving body is calculated as the rate of change in traveling distance vis-a-vis time. The calculated speed is the same as the slope which shows the instantaneous rate of change of the traveling distance over a time period. The differentiation process helps to calculate the rate of change of variable p with respect to q. The graph of q made against that of p is called the gradient of the curve. The differentiation means the ratio of a small change in a particular function with a small change in another function that relies on the first function. The larger or smaller value of a function or quantity like velocity, acceleration of moving objects, and tangent of a curve are calculated by the process of differentiation. When y is = f (x) that is differentiable then the differentiation process is expressed as f'(x) or dy / dx.

What is Derivative?

The means of derivative y = f (x) geometrically is the slope of the tangent to the curve y = f(x) at ( x, f(x)). The computation of the derivative of a given function with the help of the limits is considered the first principle of differentiation. The derivative of a function is generally represented as d/dx. Suppose a function of a specific curve be y = f(x). Let us consider the P point with its coordinates (x, f(x)) on that curve. Let us consider another Q point with its coordinates (x+h, f(x+h)) on that curve. Now PQ becomes the secant line to the curve. The slope of the curve at a specific point refers to the slope of the tangent line at that point. We know that the slope of the secant line PQ is (y2−y1) / (x2−x1).

Hence,

We need ‘h’ to be as less as possible to find the tangent slope. We know y = f(x). If the value of x is increased and denoted by Δx, then the value of y will also be increased and denoted by Δy.

So, y + Δy = f(x + Δx)

f(x) + Δy = f(x + Δx)

Δy = f(x + Δx) – f(x)

By dividing the above equation on both sides by Δx,

As the change is too small in value, applying the limits here

Here d/dx denotes the differential coefficient which is also called the Leibnitz symbol.

When the limits are applied, f'(x) is considered the first derivative of f(x). This process of finding the derivative of the given function is called differentiation.

The derivative is defined as when f is a real-valued function of a given real variable that is defined on an open interval I and if y = f(x) is a differentiable function of x then

List of Differentiation Formula

Let us suppose that a function y = x^n , n > 0 then f(x + Δx) = (x + Δx)^n and f(x + Δx) – f(x) = (x + Δx)^n – x^n

Here y = x + Δx and y → x as Δx → 0

In a similar way, we can extract the derivatives of other functions like algebraic, exponential, and trigonometric functions with the help of the basic rules of differentiation.

Differentiation Formula for Elementary Functions

The derivative formula helps to calculate the derivatives of the given functions and if the derivatives are of elementary functions then these are called the differentiation formulas.

1. The derivative of a constant function is 0: When y = k, where k is any constant term then y’ = 0
2. The derivative of a power function: When y = xn , n > 0 then y’ = n x n-1
3. The derivative of logarithmic functions: When y = ln x, then y’ = 1 / x and when y = loga x, then y’ = 1 / [(log a) x]
4. The derivative of an exponential function: When y = a x , then y’ = ax log a

Differentiation Formula for Trigonometric Functions

The concept of Trigonometry in mathematics shows the relationship between angles and sides of a given triangle. There are mainly six trigonometric ratios like sine, cosine, tangent, cotangent, secant, and cosecant. You should know the fundamental trigonometric formulas on the basis of these trigonometric ratios. Here the differentiation formulas for derivatives of trigonometric functions are given below.

Differentiation Formula for Inverse Trigonometric Functions

The inverse of trigonometric ratios is considered the Inverse trigonometry function in mathematics. Here the differentiation formulas for derivatives of inverse trigonometric functions are given below.

Differentiation Formula for Special Functions

When x= f